METU NCC Mat 120 Spring 2010 Midterm 2

# METU NCC Mat 120 Spring 2010 Midterm 2 - Ni 1:: 1 u...

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Unformatted text preview: Ni 1:: 1 u Northern Cyprus Campus Calculus for Functions of Several Variables II. Midterm Last Namei g D ﬁwl-fp. 5“ Name 3 midth No‘ Semester I Spring Department: Section: Date : 8_ 5, 2010 Signature: Time 3 9:00 8 QUESTIONS ON 6 PAGES Duration 1 120 minutes TOTAL 100 POINTS .-_ll-H 1. (4+4+4+4=16 points) Consider the points A(1, 0, D), B(0,1,0), C(0,0, 2) and D(—1, ~110) in R3, given in Cartesian coordinates. Code 1 Math 120 AcadYeari 2009—201 0 (a) Find an equation of the plane passing through A, B and C. 6 I (c) Find parametric equations for the lines BC and AD. 4 (0x4; 2) a C. (gar—to) g . is (0,933 A g/Y’fﬂl 0) (mac) (“9, 0) KG): (a h 93 t (04”*"75 "er-t:- + ’sz 2 (d) Find the distance between the lines BC and AD. ‘i’Lﬁ’ﬁ-{SL AD Ii iLa KC iaan\$ Z:- K-Zf—ljo>KCO&-ii a) =<—2 q 2> ‘ A i J I m c: 3m in MI Ia’hzlib} :1 J? 2. (18 points) Match the following quadric equations with their graphs and give their name (tag. paraboloid). (1) —3:—y2+z2m0 (ii)\$2Ay2+z2:0 (iii) wm2+y2+z2=1 ,_ z 2 E z z 2 2 '2 L217 ><~%"'8 H‘s-’4‘”? E 2 +z~i X z: ‘3 . “Fagin. 3pas-(,Lwﬁmx héuma ijgpgux a? a 5M (iv) \$2—y2+zg=—1U (v) —\$+y2v22=0 (vi)x2+z2:1 7'»:— x2+32+i E“ x: ?22"“ jwﬂ gaiw‘ig 0'? 359» 5242836: Rafa-139:}; ?=.mia{3a§ C3 1“ (vii) \$2+4y2+422=4 (viii) \$2—y2—i—22m ire; (1X) 3; my+z = *2 éi/ﬁsj eﬂipgm‘ae E? - @221 x r22, LMQW‘LG‘EOV‘g é; 6-5.9,ng (?a.m!maia-‘DE 3. (5 points) Describe the surface p2 — 4p+ 3 = O in R3, Where ,0 is the radial coordinate in the Qpﬂﬁga—D: {D [if—'31} spherical coordinate system. 4. (5 points) Find and sketch the domain of the function \$+y f(\$ay):—rm 5. (5 points) Does the following limit exist? If so, What is its value? W . \$28l1'1y LAME?- each-25: M 11m (::,;;}»(0,0) m2 + y? {3 Pi c3 ‘7 i ~ W" V269??? SEW (rsael “M. iv {M ﬂaw-ﬂ) . “up, 0 ﬂ ‘ L I; i' / \ 2. 5'; Z W i" (3,9519 Or) g?“ y T z <— 4 E W ‘ La I i/ in i g h“ - gfi Li... 15‘) W H r r—bg r We U rag) M if 6. (5+5+5=15 points) Notice that the graph of any function y = f(m) on the mywplane can be parametrized as r(t) :< 75, ﬁt) >. Suppose that f has a continuous second derivative. (3) Find r’(t) and r”(t)i F’G): 4% in “9113(0) '1 < i? BY?) > :1 Z V (4)1: 34.: i.“ in £(p)> (b) Find parametric equations of the tangent line to the curve at (to, f (120)) using the information in part (a). maﬁa-'11 {Me 5965 Kan-5L (Ii-u, 36;”: D :L atmch (é. \$6535 g 7719-:- (:30) i(€-,>> + <:, We,» :9 : <50 +2}? iéa)?1gr(ﬁ}t> ¥_ﬁ‘—b—‘_—,——I Km” 50% {3ft} Mean ﬁrm-e (c) Show that the curvature is 0 at the points of inﬂection of this graph. kg) Er’iﬂxr'fﬂi E<sF g’fz)’o>x<aﬂ”r’éﬁgo>§ + :: EV‘H) E? ii Ec grates}? : £40; ﬁat); \$"(1¢}>E g1? @Yn}zfzw '3 “WM h.—.._.._,_.—_,_.m.m_. (5+ WM“ )3" it Wow in Most; E 2 2 7. (6+6+6:18 points) Let t (132 + y1n(z —E— 1), and V m —§§ (a) Find va(1,—1,0). {ﬁg-7 (2%) thHEJ 2:; > «F Vﬁﬂg—Eio): <2: 5am): "57> 2": <2: 0} -;> __ l ,, W - <4; ﬂag erg>o<2ﬁm> :24 — 34.0 i t (b) Compute %. 51/; f __ 3 (we? (0) If 5: = 51?, y = s + t2, and z = t — 52, ﬁnd by using the Chain mm A Eéfivr ﬁin/sﬁ 5/: 3;; j“: 3 (£336?) (in (?+:))(;§ 5? (.25) //—--—:-"7 519\$ '52 4- La (f'szH) A?” i >625) f 1 1 W 8. (18 points) Let f(.r7 = \$2 + 23: + y2 + 2y. Find the absolute maximum and minimum of 3.» f on the region D defined by the inequalities 0 g T g 2, 0 S 6 3 —2I—‘ in polar coorn 23/ ﬂags: an; +25 ...
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## This note was uploaded on 03/23/2011 for the course MAT 120 taught by Professor Various during the Spring '11 term at Middle East Technical University.

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METU NCC Mat 120 Spring 2010 Midterm 2 - Ni 1:: 1 u...

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